Linear F-manifolds, a duality and the generalized tangent bundle

TL;DR

This paper systematically studies linear F-manifolds, introduces a duality between them on a vector bundle and its dual, and explores their relation to the generalized tangent bundle, with key examples like tangent and cotangent prolongations.

Contribution

It develops a systematic framework for linear F-manifolds, establishes a duality using connections, and investigates compatibility with the generalized tangent bundle.

Findings

Duality between linear F-manifolds on E and E* established.

Main examples include tangent and cotangent prolongations.

Compatibility conditions with the generalized tangent bundle are characterized.

Abstract

A linear F-manifold is an F-manifold (E, \circ , e) defined on the total space of a vector bundle \pi : E \rightarrow M for which the multiplication and unit field are linear tensor fields. We develop a systematic treatment of linear F-manifolds. Using an additional suitable connection on M, we define a duality between linear F-manifolds (with and without Euler fields) on E and the total space E^{*} of the dual vector bundle. Our main examples of linear F-manifolds are the tangent and cotangent prolongation. Motivated by the direct sum of tangent and cotangent prolongation, we define and investigate compatibility conditions between linear F-manifolds and the geometry of the generalized tangent bundle.